2 PARTICULAR CASES AND EXAMPLES 249 PROBLEMS 1. Let G be a locally compact group, A a dense subset of G, ft a left Haar measure on G, and H a /^-measurable subset of G with the following property: for each se A, the sets sH n QH and H n (J^H are ^-negligible. Show that either H or its complement is negligible (prove that the measure <pH " p is left-invariant). 2. Let G be a locally compact group, ft a left Haar measure on G, and A, B subsets of G. (a) Suppose that one of the following two conditions is satisfied: (a) A is /i-integ- rable; (ft) ju*(A) < + °o, and B is /^-measurable. Show that in either case the function /O) = fji*(sA n B) is uniformly continuous on G with respect to a right-invariant distance on G. (For any two subsets M, N of G, put p(M, N) = />((M n CN) u (N n CM)). Consider first the case where A is compact, and show that for each e > 0 there exists a neighborhood U of e in G such that p(sA n B, stA n B) <^ e for ail s e G and all / e U. Then apply Problem 5 of Section 13.9. If B is /^-measurable and /z*(A) < + oo, observe that there exists a decreasing sequence (An) of jn-integrable subsets of G con- taining A and such that inf(/x(An)) = ft-(A), and show that B) = inf(/^*(5An n B)) (Section 13.9, Problem 2(a)). Use the fact that ju,(An - An+1) tends to 0 with 1/w.) (b) If A is /z-integrable and /x*(B) < + oo , then the function / is also uniformly continuous with respect to a left-invariant distance on G. If also A"1 is /z-integrable, then f(s) <& = /x(A~I)/x*(B). (Reduce to the case where B is a-integrable, and JG observe that in that case /x(,yA n B) = p,(A n .s^B), and that <psA. n B = ^SA^B and <psA(0 = <ptA-i(.s).) (c) Deduce from (a) that in the two cases considered there, the interiors of AB and BA are nonempty if A and B are not ju-negligible. (d) In the group G = SL2(R), give an example of a compact set A and a jit-measurable set B such that the function f(s) — p,(sA. r\ B) is not uniformly continuous with respect to a left-invariant measure on G. (Observe that there exists a sequence (tn) of elements of G tending to e and a sequence (sn) of elements of G such that the sequence s^1tnsn tends to the point at infinity.) 3. Let G be a locally compact group, p, a left Haar measure on G, and A an integrable subset of G such that /i(A) > 0. Show that the set H(A) of elements s e G such that /x(A) = /x(A n sA) is a compact group. (Use Problem 2 to show that H(A) is closed in G. To show that H(A) is compact, consider a compact subset B of A such that /i(B) > i/x(A), and prove that H(A) <= BB'1.) 4. Let G be a commutative locally compact group, written additively. Let /x be a Haar measure on G and let A, B be two integrable subsets of G. (a) For each 5 e G let A' = <rf(A, B) = A u (B + s), W « rs(A, B) = (A - ,y)nB. Show that /z(A') + /4B') = /z(A) 4- MB) and that A' + B'^A + B. (Note that A + <f> = (/> for all subsets A of G.) onto L£(X, JLC) for 1 <p g + oo.